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Systems Reliability, Supportability and Availability Analysis. System Reliability Modeling and Analysis r-out-of-n and Standby Configurations. Systems Reliability Models. r-out-of-n Reliability Configuration. System Reliability Models - r-out-of-n Configuration.

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System Reliability Modeling and Analysis r-out-of-n and Standby Configurations


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systems reliability models
Systems Reliability Models

r-out-of-n Reliability Configuration

system reliability models r out of n configuration
System Reliability Models - r-out-of-n Configuration

Definition - a system containing n elements, out of which at least r are required for system success, is the so called r-out-of-n reliability configuration

Remark - the r-out-of-n reliability configuration is a general configuration. If r = 1, the configuration is a parallel configuration. If r = n, the configuration is a series onfiguration.

Example - a piece of stranded wire, with n strands, which at least r strands are necessary to support the required load

system reliability models r out of n configuration1
System Reliability Models - r-out-of-n Configuration

Reliability block diagram

Assumption - the system consists of

n identical and independent elements

Remark - the number of elements, r, surviving time t, is a

random variable with Binomial distribution

E1

E2

En

r-out-of-n

system reliability models r out of n configuration2
System Reliability Models - r-out-of-n Configuration

Element reliability Ri(t) = R(t) for i = 1, 2, ... ,n

System reliability

System failure rate

System mean time between failures

system reliability models r out of n configuration3
System Reliability Models - r-out-of-n Configuration

Exponential distributions of element time to failure

Ti ~ E() for i = 1, 2, ... 5 special case: n = 5 and r = 3

Reliability Block Diagram:

E1

E2

E3

E4

E5

3 out of 5

system reliability models r out of n configuration continued
System Reliability Models - r-out-of-n Configuration continued

System reliability

System failure rate

System mean time between failures

system reliability models r out of n reliability configuration
System Reliability Models - r-out-of-n Reliability Configuration

Element time to failure is exponential with failure rate 

n = 3 and r = 1

Reliability block diagram:

1 out of 3

E1

E2

E3

system reliability models r out of n reliability configuration1
System Reliability Models - r-out-of-n Reliability Configuration

System reliability

System failure rate

System mean time between failures

slide10

Example

The reliability configuration for a system is r-out-of-n, with r = 3 and n = 5. If T has an exponential distribution with parameter , find:

a. RS(t)

b. hS(t)

c. MTBFS

system reliability models standby configuration
System Reliability Models - Standby Configuration

Definition - the standby reliability configuration consists of one or more elements standing by to take over the system operation on occurrence of failure of the operating element

Remarks

Usually a standby configuration requires failure

sensing and switching devices to monitor the operating

element and to switch a standby element into operation

whenever a failure is sensed

The standby elements can be completely de-energized

‘cold standby’ or partially energized ‘warm standby’

system reliability models standby configuration1

E1

M

S

E2

E3

En

System Reliability Models - Standby Configuration

Reliability block diagram

where E1 is the initial operating element

E2, E3, ... En are initial standby elements

M is the failure sensing device

S is the switching device and

system reliability models standby configuration2
System Reliability Models - Standby Configuration

Assumptions:

Perfect failure sensing and switching

Zero failure rate during standby

Two identical and independent elements

Element time to failure is exponential with parameter 

Reliability block diagram

System success results if E1 survives time t or if E1 fails at

time t1, and E2 survives time t - t1

E1

E2

system reliability models standby configuration3
System Reliability Models - Standby Configuration

System reliability RS(t) = (1 + t)e-t

System failure rate hS(t) = 2t/(1 +  t)

System mean time between failures MTTFS = 2

system reliability models standby configuration4
System Reliability Models - Standby Configuration

Reliability contribution

of the 1st element

Reliability contribution

of the 2nd element

system reliability models standby configuration5
System Reliability Models - Standby Configuration

Considerations:

Perfect failure sensing and switching

Zero failure rate during standby

Independent elements

Exponential distributions of element time to failure

Ti ~ E(i) for i = 1, 2, ... ,n

System mean time between failures

system reliability models standby configuration7
System Reliability Models - Standby Configuration

If i = , i = 1, 2, ... n, then

and

system reliability models standby configuration conclusions
System Reliability Models - Standby Configuration Conclusions

As the number of redundant paths increases, the mission reliability approaches the reliability of the monitor/switching device.

When the failure rates of the path, the switching devices, and the monitor/switching device are equal, standby redundancy with two paths results in a mission reliability considerably less than that of a single non-redundant path.

For systems where the switching device and monitor failure rates are less than the path failure rate, the greatest increase in reliability occurs when one redundant path is added to a single path.

system reliability models conclusions continued
System Reliability Models - Conclusions continued

For a given path and switching device failure rate, reliability improvement increases rapidly as the monitor failure rate decreases and the number of redundant paths increases. The same is true if the monitor failure rate is held constant and the switching device failure rate decreases.

Significant improvement in mission reliability through redundancy results from the utilization of switching devices and monitors that are much more reliable than the path being switched.

configuration considerations in design
Configuration Considerations in Design

Series Configuration - Relative to Redundant Configuration

Simpler

Increases Basic Reliability

Reduces Support Resources

Decreases Mission Reliability

Redundant Configuration - Relative to Series Configuration

More Complex - Increases Weight

Requires More Testability

Increases Support Resources

Decreases Basic Reliability

Increases Mission Reliability